Explicit formulas for partition pairs and triples with 3-cores

Let A3(n) (resp. B3(n)) denote the number of partition pairs (resp. triples) of n where each partition is 3-core. By applying Ramanujan's ψ11 formula and Bailey's ψ66 formula, we find the explicit formulas for A3(n) and B3(n). Using these formulas, we confirm a conjecture of Xia and establish many arithmetic identities satisfied by A3(n) and B3(n).